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AIP conference proceedings

Decoherent Histories Approach Might be Predictive even when the Decoherence Condition Does Not Hold

Norifumi Yamada, Guillaume Adenier, Andrei Yu. Khrennikov, PEKKA J. LAHTI, Vladimir I. Man’ko, Theo M. Nieuwenhuizen

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The decoherent (or consistent) histories approach is applied to the arrival time problem. Since the decoherence condition does not hold for arrival time, a probability distribution of arrival times cannot be defined. Instead, a quasidistribution, which is not positive definite, can be defined in the histories approach. It is shown that well‐known Kijowski's distribution, which is positive definite and often claimed to be the correct distribution of arrival times, is obtained from the quasidistribution when the momentum wave function has certain properties. This suggests that the histories approach still holds some predictive power even when the decoherence condition does not hold.

DOI
10.1063/1.2827332

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The decoherent (or consistent) histories approach is applied to the arrival time problem. Since the decoherence condition does not hold for arrival time, a probability distribution of arrival times cannot be defined. Instead, a quasidistribution, which is not positive definite, can be defined in the histories approach. It is shown that well‐known Kijowski's distribution, which is positive definite and often claimed to be the correct distribution of arrival times, is obtained from the quasidistribution when the momentum wave function has certain properties. This suggests that the histories approach still holds some predictive power even when the decoherence condition does not hold.
In this paper we study the construction of probability densities for time of arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time of arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of the Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit—there are, however, problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
Decoherent histories approach to quantum mechanics assigns probabilities to events that are not restricted to a single moment of time, provided that the decoherence condition holds for the set of events. The condition is however hardly satisfied in general, which has been considered to limit the usefulness of the histories approach. The present paper applies the histories approach to the tunneling time problem to show that, although the decoherence condition does not hold, the histories approach is still useful in predicting the central tendency and the dispersion of resident times. We consider such quantities that would become the average and the standard deviation of resident times if the decoherence condition held. Since the condition does not actually hold, they are not regarded as the true average and the true standard deviation. Nevertheless, a quantum clock approach allows us to interpret these quantities as a central value and a dispersion of resident times within the time resolution of the clock.
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit–there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the timeof-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski. 1
We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of the Lindblad form. We specialize to the case of quantum Brownian motion, and show that after a time of order the localization time of the current becomes positive. We show that the arrival time probabilities can then be written in terms of a positive operator-valued measure (POVM), which we compute. We perform a decoherent histories analysis including the effects of the environment and show that time-of-arrival probabilities are decoherent for a generic state after a time much greater than the localization time, but that there is a fundamental limitation on the accuracy $\ensuremath{\delta}t$, with which they can be specified which obeys $E\ensuremath{\delta}t\ensuremath{\gg}\ensuremath{\hbar}$. We confirm that the arrival time probabilities computed in this way agree with those computed via the current, provided there is decoherence. We thus find that the decoherent histories formulation of quantum mechanics provides a consistent explanation for the emergence of the probability current as the classical arrival time distribution, and a systematic rule for deciding when probabilities may be assigned.
In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator T(X)\mathcal{T}^(X)T(X) whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive-definite operator J(+)(X){J}^{(+)}(X)J(+)(X), which is nothing but a straightforward quantum version of the modulus of the classical current. For quantum states highly localized in momentum space about a certain momentum ${p}_{0}\ensuremath{\ne}0$, the expectation value of J(+)(X){J}^{(+)}(X)J(+)(X) becomes indistinguishable from the quantum probability current. This fact may provide a justification for the common practice of using the latter quantity as a probability distribution of arrival times.